On the square integrability of the q-Hermite functions

Overlap integrals over the full real line −∞<x<∞ for a family of the q-Hermite functions Hn(sinκx¦q)e−x22, 0<q=e−2κ2<1 are evaluated. In particular, an explicit form of the squared norms for these q-extensions of the Hermite functions (or the wave functions of the linear harmonic oscillator in quantum mechanics) is obtained. The classical Fourier-Gauss transform connects the q-Hermite functions with different values 0<q<1 and q>1 of the parameter q. An explicit expansion of the q-Hermite polynomials Hn(sinκx¦q) in terms of the Hermite polynomials Hn(x) emerges as a by-product.